Some Examples related to the 4/9/02 lecture on polar graphs.

Some of you raised interesting questions about the effect of minus signs and square roots (etc) on basic polar graphs. I hope these examples help clear those up, though there are surely more questions that you can ask, and I hope you will explore some more on your own. I created these with MathGV (see my help page for the link to this freeware).

Notes: 1) Please ignore the little rectangles accidentally appearing near some of the graphs. 2) I printed equations in the same color as their graphs - two equations of the same color have the same graph. 3) y^2 means "y squared" and cos^2(4t) means "cos(4t) squared".

Here are some familiar graphs. The thin dark blue line is y = x. If you square both sides (y^2 = x^2), you include the possibility of y = -x. So, you "create new solutions" and get a bigger graph (the red "X"). On the other hand if you take square roots of both sides of y = x, you "destroy some solutions" (you prevent negative x's and y's) and get a smaller graph (the green line).

NOTE: Some of these lines should be on top of each other. I had to play with the widths of the lines (etc) so you can see them.

NOTE: What happens if you take the square root of both sides of y^2 = x^2?

This is a polar version of the previous example. The original graph is the red one; r = cos(4t). Squaring both sides does not create new points this time [because -r = cos(4t) has the same graph as r = cos(4t). The next example explains this more]. Taking a square root of cos(4t) [the blue graph] does two things:

• It destroys some solutions, such as when t = pi/4 [we can't take sqrt(-1) etc] and
• It distorts the four remaining petals. [they get bigger, because (for example) sqrt(1/4)>1/4 etc]

The blue graph, r = cos(t) is familiar. Putting a minus sign on the r has the effect of rotating every point 180 degrees around the origin, giving us the red graph. We get the same effect by adding pi to the angle (or subtracting pi from it). In fact, cos(pi+t) = - cos(t). This changes graphs like r = cos(3t) but not graphs like r = cos(4t).

NOTE: r = sqrt(cos(t)) looks about the same as the blue circle; some solutions are destroyed, but because of retracing, you don't really notice this.

This example combines the ideas above. The original graph is the green limacon; r = 1.5 + cos(t). Changing to "-cos(t)" rotates the graph (as discussed above) and gives the red graph. Multiplying either side by -1 rotates the graph again, taking us back to the green graph.This explains why -1.5 + cos(t) has the same shape as +1.5 + cos(t).

NOTE: Practice drawing a lot of graphs like these using pencil, paper [and maybe a non-graphing calculator]. You will learn shortcuts and important patterns this way, better than just reading/hearing about them. Later, you can use graphing calculators and/or computers for more speed and accuracy.